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RÉSUMÉS / SLIDES
Brice HALIMI (Univ. Paris Ouest, IRePh) My talk will bear on the particular ways in which a mathematical object is introduced and cognitively handled. Focusing on a particular basic example, taken from combinatorics, I will set out a possible misunderstanding about it, which, albeit clearly wrong, is not senseless. My aim is two-pronged. Firstly, I will reconsider the "identity problem" faced by Shapiro's ante rem structuralism, and endeavor to show that this problem comes from giving too little attention to settings as an essential component of mathematical objectivity. Secondly, I will argue that some misunderstandings in mathematics can be understood as misunderstandings about mathematical settings, and that, in contrast, such misunderstandings cast some light on mathematical understanding itself.
Analytic number theory uses real and complex analysis to study problems about the prime numbers. In the 18th century Euler uses analysis to re-prove the infinity of the prime numbers, and his approach then inspires Dirichlet and Riemann in the 19th century. This path of research leads to important results about the distribution of primes in the work of Hadamard and de la Vallé-Poussin around 1900, because the reduction of questions about integers to questions about the divergence and convergence of certain series, offers much more powerful and flexible techniques than algebra in many cases. Conversely, once this habit of transposing problems upstairs to real and complex analysis is established, problems that arise originally in the infinitesimal calculus turn out to have important consequences for the study of the integers: the study of elliptic functions begins at the end of the seventeenth century in connection with the mathematical modeling of the pendulum, which entails finding a way to determine the arc length of an ellipse. The eighteenth century tendency to study problems of number theory analytically, embedding the study of the integers in the study of real-valued functions, and the nineteenth century tendency to embed real analysis in complex analysis, provides an important background for understanding the reduction of Fermat’s Last Theorem to the Taniyama-Shimura Conjecture. Important problem-reductions combine, juxtapose and even superpose discourses that are more concerned with analysis, and discourses that are more concerned with reference. Wiles’ proof is not only about the integers and rational numbers; it is at the same time concerned with much more ‘abstract’ and indeed somewhat ambiguous and polyvalent objects, elliptic curves and modular forms. So for example at the culmination of Wiles’ proof, where analysis has invoked cohomology theory, L-theory, representation theory, and the machinery of deformation theory, we find the mathematician also involved in quite a bit of down-to-earth number-crunching. (Wiles 1995) I argue that this polysemy plays a useful role in the proof, and throws interesting light on the objectivity of the things of mathematics, as well as the growth of mathematical knowledge. Ralf KRÖMER (Univ. Wuppertal) The first part of the talk briefly recalls the debate on the lack of set-theoretical foundations for category theory, and its philosophical significance, especially focusing on suggestions made in my 2007 book - where I discussed what it does mean for a category to "contain" itself as one of its objects (when translating set-theoretical propositions to category-theoretic ones in Lawvere's way), and the grounds on which workers in the field felt justified in using various category-theoretic constructions (and whether or not this feeling is related to a hoped-for possibility to reduce the constructions to trusted-in set-theoretical axioms).
Brendan LARVOR
Marianna ANTONUTTI MARFORI (IHPST) & Benedict EASTAUGH (Univ. of Aberdeen)
Julien PAGE (CNRS, ERC PhiloQuantum & SPHERE)
Alejandro PEREZ CARBALLO (Univ. Massachusetts Amherst) Many, if not all, of the metaphysical and epistemological concerns that arise for varieties of mathematical realism arise, mutatis mutandis, for varieties of metaethical realism. It is thus somewhat surprising that a family of views in meta-ethics that has emerged as a response to such concerns (varieties of *expressivism*, *non-cognitivism*, or *quasi-realism*) does not seem to have an official analog in the philosophy of mathematics. What could such a view look like? Is it at all plausible? I consider a number of reasons why one might think such a position cannot get off the ground and find them all wanting. I then suggest that, as a thesis about the role and nature of mathematical thought and talk, expressivism is a natural ally of versions of 'thin-realism' advocated by Penelope Maddy, John Steel, and Bill Tait, among others.
Giorgio VENTURI (Unicamp, IHPST In this presentation we try to elucidate the notion of natural axiom in set theory. We will review the dichotomy of intrinsic-extrinsic justifications and we will find both theoretical and practical difficulties in their use. We will describe the limits of the theoretical framework, that we will call conceptual realism, where the intrinsic-extrinsic dichotomy is usually placed. In outlining our view of naturalness in mathematics, we will argue that an axiom is to be considered natural with respect to the fundamental ideas that motivate the formalization of a theory. In the case of set theory: the clarification of the notion of arbitrary set. In the end, after a brief presentation of Forcing Axioms, we will argue in favor of their naturalness as a case study for our proposal.
Sean WALSH (Department of Logic and Philosophy of Science, University of California, Irvine) This talk sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church's intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, we discuss a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms of Church's intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin's intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions.
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